Wood filler- sold at home depot or any similar store. Then you can stain it or paint over it. I can't live without it in this old house. It dries super fast though, so if you use it, cover up the parts you don't want caked in wood filler with painter's tape.

http://www.homedepot.com/h_d1/N-5yc1v/R-100376258/h_d2/ProductDisplay?catalogId=10053&langId=-1&keyword=wood+filler&storeId=10051

Don't use wood filler. Wood expands and contracts with the seasons. Wood filler does not. Wood filler is hard. As the wood expands (in the more humid summer months) the boards press against the fill. As the wood presses against the wood fill, the wood fill dents the boards, compressing the fibers. Then winter comes and the wood shrinks. Now the wood has been compressed from the wood fill and it is naturally shrinking as a result of the low humidity and dry heat of indoor heating. The wood fill falls out.

Some suggest using foam backing rod and paintable caulking. Here's a tutorial.

http://www.oldhousejournal.com/Restoring_Painted_Wood_Floors/magazine/1074

P.S. I've done this before and I was surprised at how much the wood expanded and pushed out the caulking. I had to use razor blade to cut off the caulking in the summer months. That was a hassle and it was difficult to do without damaging the floors. So sse less caulking than you think is necessary. When your done the caulking in the gaps should be concave, not flush. It is best to do this in spring or fall, when boards are in mid expansion/contraction.

p.s. http://www.mcilvain.com/wood-moves-get-over-it/

please excuse typos.

That is an extremely rustic floor. Where are you located? I've used wood filler for dents in my pine floor but not huge gaps between the boards. Can't you get the landlord working on this rather than the pregnant lady.

A trick master wood workers use is to mix sawdust from the same kind of wood with wood glue and fill cracks with that. I think, however, that your gaps are too big. Can you find a master cabinet maker to ask? That's who would know.

Try this, for one:
https://www.google.com/search?sourceid=chrome&ie=UTF-8&q=master+cabinetmaker

I agree with not utilizing wood filler due to the lack of expansion property where this product is concerned. I now utilize bathroom caulking instead of fillers of any type on cracks in walls or small ones in a painted wood floor we have in one of our rooms. Bathroom caulking will expand and contract somewhat. Application is a little tricky as you have to smooth the area well with a putty knife otherwise once painted is is readily apparent where the caulking was applied.

We had the same issue in our old house. Our floors are in fairly good shape but old and a lot of gaps in certain areas. We also did not want to refinish it because of the cost and we have two small children in the house. I scoured the internet for solutions on this issue and all the same solutions as the previous posts but I did come across a different one I believe it was a this old house solution that they may have used in the old days, who knows, they would get rope and push it into the cracks and then just stain it the same color as the floor. I guess it's a quick fix.

We have the same sort of problem on the second floor of our 100 year old house (first floor is hard wood, no problems, but second floor was done in cheaper pine, which now has lots of gaps and cracks). The prior owner tried a lot of the wood putty and wood glue/sawdust type options, and these have all popped out over time. We cover it with rugs or furniture as much as possible, in the bad spots, and plan for some theoretical day when we will likely put in a whole new wood floor there, which I think is really the best option despite how much I like a rustic look and original floors, and all that. We did have a contractor come in and replace some of the more dangerous/prominent spots with "new" pieces of old flooring, sourced from a local renovation reuse shop. They had to restain those sections, and it is not a perfect match to the floor, but it was an improvement. If you have any really bad areas, your landlord should consider doing these kinds of patches. We had nails that would pop up repeatedly and rip your sock (or foot), etc., and with little kids in the house, it just wasn't acceptable.

If you want to have a little fun with it, you could try something along the lines of this:
http://www.janvormann.com/testbild/dispatchwork/

Wood shrinks over time as the moisture leaves it. That creates the gaps in between the floor boards. This Old House recommends filling those gaps with hemp after sanding before staining and coating. It allows for that expansion and shrinking due to tempurature changes.
It looks like there's more going on with your floors than shrinking so you need to first check for termites and dry rot and treat if necessary. Also check the distance between your floor joists. Some older homes have floor joist 24" on center which was the norm way back when. Today it's 16". If yours are 24", there is too much weight on the wood and it's causing the wood to bow between the floor joists and pop out where the ends of the planks are attached to the joist. Sadly, the only solution to curing this is to take up the floor (it can be done!) install additional floor joists in between the existing ones, and return the floor to it's rightful place. The only problem with this solution is that there will be some lost flooring. It's old, it's dry, it's brittle. It will crack. Also, if it is tongue and groove (most is, and you can confirm if your is or isn't by a seemingly lack of nails-the nails are actually hidden in the tongue part). You will lose at least one plank due to taking up the first piece and possibly more due to the age of the wood.
Hopefully 24" floor joists are not the cause and it's simply some localized damage which is easily dealt with by replacing the damaged planks. Whatever the cause, there are many resources for replacement wood: take it from inside the closets or from a another room in which you replace the entire floor, freecycle.com, the 'Free' section of Craigslist, and of course, wood recyclers.
Whatever your solution, good luck with it and congratulations on the baby!

BTW, if your floors are like most floors put down during the 1880s, they're nice and thick, probably 5/8" of an inch, possibly more. This makes them able to be refinished many times, unlike new flooring of today that has only an 1/8" of real wood glued to pressed wood. Spending the time to refinish them would be worth the trouble. After refinishing, if left unstained, they'll turn slightly darker as they are exposed to light. My 1883 pine floors were refinished about ten years ago. They started out a pale honey color and today they are a warm, comforting almond color and I couldn't be happier.

I used to work at an Anthropologie that had all reclaimed barnwood flooring and was FULL of holes. We covered the large ones with copper sheet metal and it looked awesome )and prevented our heels from getting stuck). You can just use it the way you are using the duct tape now.

http://i237.photobucket.com/albums/ff16/darvis73/Bodie/DSCN2786.jpg

Your floors are beautiful. I'm not sure how this would look on white-painted floors but my favorite cafe has very old (unpainted) wood floors as well. They have patched similar holes with copper sheeting (I think it is the stuff used around fireplaces etc.). They just cut a square to size and then hammer it on with tiny copper nails. It looks great... natural, raw and interesting in a creative way. Good luck...and may the birth of your little one go perfectly!

The only reason I would suggest this is because, if that picture actually is your floor, it is really old, really beat up and most importantly, not yours. We did this with a rental property with a very ugly deck. My husband cleaned it first, then took a gallon of good deck paint and a long handled roller and just painted it again.