I have these super ultra-cheap bike tires that have the following label on them;
700x250 , (25-622) , 28x1-5/8x1

I can't find this size anywhere. They were HELL to get on my rims. My rims don't have a size, but I'm guessing they are 700c. They are labelled Aofeng 8081TS

I might just buckle and buy new rims, tubes, and tires - but I'd rather not. My tubes are flat and I need new ones. What size should I get for this odd tire size?

Yep, 700c. 25mm. Some road tires are REALLY hard to get on some road rims.

Tubes will probably be a range. 25-28mm or something like that.

The first label is probably actually 700x25C (no idea why there's the penchant for putting the 'C' next to the width rather as 700c).
The second label is consistent with this, giving the nominal dimension in mm: 25mm width, 622mm bead seat diameter of the rim - aka '700c'.
The third label is inconsistent. Sometimes 700c size wheels and tires are referred to as either 28" or 29", but usually this is when wider tires are used. And if the tire width is about 25mm then it would be about 1", not 1 5/8".

Just about any reasonably narrow 700c (or 27") tube should work fine. They're usually labeled with a range of widths since they stretch so easily. Something like 19 - 28mm. Make sure you get the right valve style (Shrader or Presta) to match the hole in the rim.

More than just the answer to your question: https://sheldonbrown.com/tire-sizing.html

As others have said, your tire is a pretty common 700 x 25C. Any 700C tire will fit the rim, with some gotchas. Read about "width compatibility" here: https://www.sheldonbrown.com/tire-sizing.html#width

Narrow tires (like yours) on narrow rims (like yours) are typically tough to mount and remove. You can make the job easier by using a thinner rims strip (this covers the spoke holes on the rim), such as Rox: https://www.universalcycles.com/shopp...1&category=140

Is it remotely possible that you actually have (nominal) 27" rims? It would only be if you have an older, most likely English, bike. 27" wheels are a bit larger than 700c.

WRT the third label, where it says "28x1-5/8x1", I don't think that's actually inconsistent; it's just because in some places the name of the size we call "700c" is "28-inch". And yes, that size is smaller than what we call "27-inch", and has the same bead seat diameter as what we call a "29er". Incidentally, 650C and 650B have different bead seat diameters; also, there are two different bead seat diameters that are both known as 20" wheels, but one is common for fatter tires and the other for skinnier. What could possibly be confusing about that???
It's sort of ironic, actually; I lived in Germany for a couple years, and over there, they call standard road wheels 28-inch. And then make fun of us for not using the metric system!
Part of the inconsistency in tire size nomenclature is because at various times they used to measure the entire diameter of the mounted and inflated tire, and use A, B, C, etc. to denote the width. Hence 700C, 650B, etc. As stuff got more standard, people got more interested in the bead seat diameter, which is obviously smaller than the overall diameter of the mounted tire. That's where the 622 part comes from.

I'm just waiting for pizza places to get in on this game. For the longest time, my better half was on a quest for a 700c pizza.