I've been playing bass for over 40 years. Forget the Squiers and Epiphones. The best cheap basses are made by Rondo:
http://www.rondomusic.com/bassguitars4.html
They also have an Ebay store, which is how I bought mine. Paid $110 over 10 years ago and the price hasn't gone up much since then. (Tried to link a photo of my bass but doesn't show up - it's a Fender Jazz copy in turquoise with a rosewood neck)
I bought mine when they had the original headstock design similar to Fender and apparently there was a copyright issue and they had to change it to the current ugly headstock. (But I see at least one bass on their website with the old headstock so I don't know what the deal is)
This bass feels more solid and not cheap like a Squier or Epi. It gets rave reviews on the TalkBass forum. They have medium and short scale basses and also 3/4 size basses.