Are Sump pumps necessary?
Sump pumps are generally used to collect water that penetrates the walls or floor of your basement (concrete is porous). It either pumps the water out to the sewer (may not be code anymore) or simply outside where it can run away from your house, or is stored in a dry well to evaporate/soak. Whether or not you need one depends a lot on how your house was built, the local water table, and the ground surrounding your house. But let me give you one general principle to start with in our discussion here:
If you keep the water out, you probably don't need a sump pump
A properly built basement will have good weeping tiles and a waterproofing membrane on the outside of your basement walls. Sadly, in older houses, weeping tiles are literally tiles and they can break or clog over time. And too many builders install too little waterproofing (or none at all). The key here is you don't want to install a sump pump to handle a problem that is outside your house. If you keep the water out in the first place then you shouldn't have a problem. So I would bring in a waterproofing specialist first to make sure you have

Sufficient weeping tiles that are not clogged/broken. This means excavating the footers
Enough waterproofing on your outside basement walls (and proper basins for any basement windows)
Sufficient surface drainage. You should have gutters to catch rain water and send it to the ground where it is channeled away. Ground settles over time and can sometimes trap water by the foundation

What if the water still gets in?
Once you're sure the outside is doing its job, then you can look inside. It's even possible the outside may need to drain inside and use a sump pump to push the water out anyways. Most likely, however, if you still have water intruding, it's due to a water table issue. The water in the ground is seeping up into the foundation. If this is the case, the solution is usually to excavate the basement and install what basically amounts to internal weepers that can catch the water and give it a path to the basin where the sump pump can remove it.
Can I do this myself?
Probably not. Repairing or replacing a pump is one thing, but installing the parts to get the water to the pump is what will likely overwhelm a DIYer. You really want a reliable expert to make sure the outside is taken care of if you have a water intrusion issue. Only then can you turn to the inside. You also need experts to look at the water table and drainage.