As this is self-hosted, I feel the majority will reccommend against a Synology or similar pre-built, closed source solution.
At a budget of 150€ it will probably be challenging to build a whole system with new parts, but it can certainly be achieved with two or three generations old, second hand hardware.
The easiest way would probably be to keep a lookout for old office midi-towers (Dell Optiplexes and similar). Those usually have a few pci slots to throw an hba into and hook up a few hard drives. The mounting of the harddrives itself will need to be handled with uhm... Creative solutions. Depending on the system, you'll probably want to upgrade the ram. And if you want easier Hardware handling, you may be able to just throw the system in a different case later on.
Another solution, and maybe even cheaper would be an old NUC or other mini-PCs. To be honest, I have no idea how people manage to use those as NAS or how you are supposed to manage multiple hard drives with them. External enclosures?
Then there's also more Pis and other micro PCs. Same challenges.
So, this writeup has not actually adressed your question: what's the /best/ solution?
I also have no idea. It really depends on what you want, what your budget is, how much you want to fidget around. How much space do you have to put a system? What is on offer around you? Does the company/university/school you work at maybe offer hardware they would otherwise need to dispose of? Check craigslist/marketplaces/ebay.
I am partial to the midi tower approach, as it offers a good deal of flexibility, depending on the included motherboard.
Hope I could offer at least some help :)
//Edith: The least energy consumption would probably be the Pi, but depending on how much HDDs you add, this will also depend on what management System you run and what HDDs you use (some NAS drives come with some powersaving features). If you are in any position to do so, talk to your landlord or Eigentümergemeinschaft and get a Balkonkraftwerk. Those 800Watts will more than offset your Homelab needs.
If you don't have enough sata ports on the mobo (the optiplex 7010 has 1x Sata), you'll need a pci sata controller, is my understanding. Not sure what other possibilities there are to connect more hdds...