Let's say I want to implement storage server, which would be used as a place to store files, images, etc. from different websites. Something like S3, but only for my projects.
I thought about some API/Gateway on PHP, which would save files from those websites to appropriate server, but is it a good way? And should I use Webdav or maybe NFS/SMB, which protocol is more secure and fast?
Can you please give me advice how to create my own storage server? Especially I want to hear about appropriate stack for that, thank you.
There are a number of projects for building your own NAS or SAN (I think that is what you're looking for). Look at the FreeNAS project for example. It does require quite a bit of memory though (depending on the size of your storage and the demands you put on it). When you want to build your own NAS, you will not need very powerful CPU's, unless you want to run apps on the NAS (FreeNAS provides a system for runnning containerized applications on the NAS, using it's storage) but you will need memory and of course plenty of disks, again depending on what your exact requirements are.
However, if you're simply looking for a place to store your own files and they are not extremely large or a huge number of files, then you could simply build a Linux server and push (or pull) the files using SFTP, it only uses OpenSSH and a single port, fully encrypted with minimal overhead.