Note that the system requirements for this release of Bedrock Linux (1.0alpha3 Bosco) have not changed from either prior release.
Bedrock Linux itself is quite minimal and will run on very limited hardware. However, the client software Bedrock Linux will use will most likely require significantly higher system requirements. See the documentation for the specific client distributions you wish to use.
Bosco has been tested on (32-bit) x86 and (64-bit) x86-64 systems. In theory it should be possible to set it up on non-x86 architecture with only minor changes - such as swapping out the bootloader - being necessary, but no effort was put into ensuring this was an option and no testing was done to actually check that this is possible.
Prior releases have been show to run smoothly on both a 1.6GHz Intel Atom and 800MHz Intel Celeron-M (ie, two different ASUS Eee PCs), and it is expected that Bosco should only improve upon this. Presumably significantly slower CPUs will also suffice. Remember, though, some client software will likely have higher requirements to run adequately.
Prior releases have been shown to run without trouble on 512MB of RAM with huge amounts of room to spare, and Bosco should only improve upon this. Much smaller amounts of RAM would likely work fine for the core Bedrock Linux software, but higher amounts will likely be expected by some client software you may choose to run.
Depending on what you choose to compile into the Linux Kernel and Busybox, Bosco fit snuggly within only a handful of megabytes of disk space (although a fully-loaded Linux Kernel can take tens if not hundreds of megabytes). However, Bedrock is nearly useless without any clients, and the clients can take quite a bit of space. See the disk space requirements for the client Linux distributions you are interested in. As a rule of thumb, a gigabyte of disk space should be set aside for each client distribution, with more preferred.