Is there any way to check if a floor is completely level without having any advanced tools?
I want to be sure that a bookcase would not need extra support
The common tool to do this is a level. One that is 4 feet long is better because it can check the floor over a wider area for flatness and level. You really only need to check the area where you intend to set the book shelf and not the whole room.
A level like this would not be considered an advanced tool as they are rather simple devices. If you find the longer sized levels too expensive you can instead find cheap short ones that you can use on the top edge of a 4 foot long, straight and even pine board.
If you want to avoid buying any tool at all you could consider the idea to use pine board like the above example but substitute the cheap level with a Smart Phone running an app that simulates the bubble style level. For example there are many apps like this in the Google Play store for Android phones. (Similar apps on iPhone at the iTunes store site).
The smart phone apps require you to calibrate your phone on a known level surface. Some folks may consider this idea a gimmick but it may be just what you need for a quick and simple check. The quality and repeatability of the cell phone type of app can vary from app to app and be affected by the quality of sensors in the phone. So if you go this direction I encourage you to download three or four of the free apps and find the one that best suits your needs.
If you have a length of clear tubing, you can attach each end to a stick that you attach to some base so it's vertical.
Fill the tube with colored water until it's about 1/2 way up the stick, then mark the sticks where the water line is.
If you move the ends around, and find a place where the water level isn't at the line, it's not level.
If your bookcase is perfectly square†, you can use a plumb bob.
You can make a plumb bob out of a thin piece of string and any suitably heavy small object tied to the end. Once you've put the bookcase into position, suspend the weight to the left/right of it - the gap between the string and the bookcase should be the same distance all the way from top to bottom.
This method should only really be used where you need it to be approximately level, since you're essentially just eyeballing it with a little extra help. But it's quick and can be quickly assembled with stuff you should already have lying around the house.
† If the bookcase is an irregular shape or wasn't assembled with perfect right angles between the base and sides then this method won't work.