The short answer is no.
The dimensions of an ATX power supply are 150 mm width, 86 mm height, and typically 140 mm depth, although the depth can vary from brand to brand. ATX PSUs also use standard mounting screw hole placement, fan placement, and cable exits, so they can be used in any case.
Server PSUs do not conform to standards. Since rack mounted servers internals change often, PSU are often designed for a specific line of servers for that model's run. This is all assuming the power connectors arent proprietary and match the ATX standard.
Your server PSU may, or may not physically fit in your desktop. Not to mention, it will most definitively not have a method to attach it to the case properly. And properly attaching it to the case is important for grounding purposes. An improperly grounded PSU is dangerous to you and your computer. Server PSUs use proprietary rails and clips, not screws, for mounting in a case. This allows them to be removed without opening the server's case. Heck, not all server PSUs have power cables, but slot/tab connectors. However, I assume yours has cables, or you wouldnt be asking. This is also assuming the cables are long enough to reach your motherboards power socket.
In addition to that, 350 W is fairly low power for a gaming PC. Depending on your CPU, GPU, and other hardware, 350 W might not be enough. You might need a more powerful PSU. Not to mention, even if the PSU is ATX pin-out compliant, it wont have the extra power connectors for a modern gaming graphics card.
Server PSUs, especially 1U, use very small fans. This means they spin very fast and are often very loud. Since, every server is different, the PSU might not blow cool air where the motherboard is needing it. This can cause overheating and even damage your computer.
In the end, you are much better off buying an ATX power supply. It will be safer, easier, and more reliable.
