New home owner here. I have never lived in a home with one of these systems, so this is new to me and I might be explaining it wrong.

I have what appears to be a Sewage Ejector System in my basement.

It looks like a black tub sunk into the floor in the unfinished part of the basement, with 2 PVC pipes coming out of it and leading into a larger black PVC pipe that appears to go into the wall (and out to the sewer I am guessing). The tub is about 20-24" across, and I am not sure how deep. There is a power cable attached to a float or something that I can see through an opening in the top of the tank.

I noticed that the toilets were flushing in a weird way, and some of the sinks were draining slowly, so I began to investigate. That's when I saw some water around this tank, like it had overflowed slightly. It also doesn't smell too great, and before this, I have never noticed a smell in the basement at all, despite the opening in the lid of the tank. I also noticed that the pump does not appear to be turning on, when normally any time I ran a sink or flushed the toilet in the basement that I could hear it turn on.

Is this something I can fix myself? Searching around on the internet and this site it looks like a likely cause might be that something is stuck IN the pump itself. Seeing as I have a daughter that flushes things indiscriminately and uses way too much toilet paper, this seems even more likely. However, I am concerned that I will be dealing with "raw sewage" by taking this thing apart. Would it be easier to have a plumber come out and take a look?

Every time you deal with Sewer Ejector be ready for an unpleasant job. What is happening in your case is one of the three things: There is some paper wrapped around the float preventing it from activating the pump. You may also have some debris stuck in the rotor disallowing it from turning. Third, the pump itself is burnt.

First, check ELECTRICAL: Usually, pump is plugged into GFCI outlet. So, first, make sure the outlet is not tripped. Also, pump should have its dedicated elec.circuit. Go to the main elec. Panel and make sure the main breaker is ON. With breaker ON, take some small elec. Appliance and plug it into the pump's elec. Outlet to TEST it - just to make sure it is getting power.

If find all is OK with electrical, than, UNPLUG the pump and remove lid of the Sewer Ejector. There are 2 pipe exiting sewer ejector: one is Vent, the other is drain. Vent terminates at the lid level. It is empty. Drain is attached to the pump. There should be a check valve and coupling. Loosen the coupling BELOW the check valve and remove it. Now, pump is ready to be removed. Lift the pump, take it outside and inspect for obstruction. Take garden hose and thoroughly clean the pump. I would suggest that you put the pump inside a bucket full of water and run it for few seconds on extension cord to make sure it works. If it doesn't, get new pump. If it does work, reinstall it.

Milo Gave you excellent advice. But if you're "squeamish" about in sewerage then your best bet would to call in outside help. Good luck, Tom