21 Inc., a bitcoin startup backed by venture-capital firm Andreessen Horowitz, is releasing its first product, a computer that seeks to make bitcoin a standard feature of web services and encourage developers to build products that integrate the digital currency.

The 21 Bitcoin Computer, which goes on sale Monday, is a small, bare bones, Linux-based piece of hardware in which the bitcoin protocol is a feature of the operating system. Any products or services built with it – games or music or any online content – would have...