It's not that hard to find great bamboo sheets that are well worth the money. You probably don't want to spend $20 on a set, but it's totally not unreasonable to spend anywhere from $40 to $250 on a solid set of bamboo sheets.
About the minimum we would spend on these would be $40. Cosy House Collection has some decent sets at around this price point. The elastic is tight and will help the sheets stay on the bed better than most other brands. They also come with a 100% money back, lifetime satisfaction guarantee. This is a company you can do solid business with.
Cariloha is another brand that, while expensive, is great to do business with. Their sheets cost around $250 for a set, but you can't go wrong choosing them. Their seams are box-stitched and are great in terms of softness and durability.
Linenwalas make a good middle of the road set at around $80. At 300 thread count, they're within decent range of other high quality bed sheets that are available.
A good rule of thumb is to stick to thread counts between 200 minimum and 500 maximum if you want quality sheets at a good price. Thread count is often artificially inflated with inferior materials to make a set seem better than it actually is. If the count is too high, don't waste your money.