Counterpoint: If you like to putter around with mechanical stuff, you can get older bikes for a decent price and fix them up and/or upgrade them over time. I got a pretty sweet Raleigh Super Course for $50 (kind of... it's a long story) at a pawn shop. Old technology, but putting a ramped freewheel on it greatly helped the shifting performance. Garage sales, pawn shops, Craigslist, and even the city dump and dumpsters can yield some decent bikes.
If you want the best of everything, this isn't a cheap hobby. If you're content with perfectly functional 20-30 year old technology, there are some great bargains to be had.
Some things to look for in a used bike:
If it's pre-1990s, looked for lugged frames, forged drop-outs, 3-piece cranks; stay away from stem-mounted shifters, stamped dropouts, one-piece cranks. You're an inch shorter than I am, so in sizing, look for a minimum of 24" or 60 cm frame sizes. If it's 1990s and up, there are some decent welded frames, especially on mountain bikes/hybrids.
This bike is a 25" frame; it fits me pretty well. It might work for you, or perhaps be just a bit too large:
This is a 24" frame; it would probably fit you well:
The distinguishing feature for me is the length of the head tube. You can see the proportions of the head tube relative to the rest of the bike. On the 25" frame, the head tube looks almost too large proportionally relative to the rest of the bike. The head tube on a 24" has a little more "normal" looking proportions. (The 24" framed bike was the one I got at the pawn shop for $50; I traded it straight up for the slightly larger 25" frame bike.)
For comparison, this is a 23" frame; it would probably be too small for you (but you might make it work):
