allows my mail clients to connect via IMAP to view and search emails
dovecot will be able to handle this part. This is what I use as a mail archive (once a year, archive all mail from the previous year from various mailboxes to my self-hosted dovecot instance). I wrote this ansible role for it.
downloads new emails via IMAP
As others recommended, imapsync should be able to handle that part.
downloads new emails via IMAP
These tools are simple enough to install and manage (one package, one config file), Docker is not needed. If you really need it to fit into your docker-based setup, build and maintain your own images.
What's your existing setup? For such a simple task, check if any of the tools you use currently can be adapted (simple text files on a web server? File sharing like Nextcloud and text files? Pastebin-like? Wiki? ...). Otherwise a simple Shaarli instance could do the trick (just post "notes" aka. bookmarks without an URL). I use this theme to make it nicer. Or maybe a static site generator/blog.
I would never recommend Odoo anymore, given how painful it is to upgrade from a major version to another. Their answer to it is basically "yeah, some complex migrations need to be done, just send us a copy of your database with highly sensitive company data, pay us to do the migration and we'll send it back to you". Yeah, lol, no.
I agree that desktop/ATX tower PCs are the most useful form factor, you can stuff all your old junk hardware in there and offer it a second life without much investment.
However with current electricity prices buying more power efficient hardware can be a better medium-term investment. 1kWh bills at 0.2516€ currently where I'm at (~EU average price), assuming an average power consumption of 50W this gives you (50×24×365)/1000×0.2516=110€/year. At this rate a 200€ investment in hardware would pay for itself in 2-3 years.
Buying a <100€ setup is not worth it for general purpose servers in my opinion, it will either be underpowered or power hungry.
My current solution is to to run all my services in KVM (libvirt) VMs on my beefy desktop computer which is already on most of the time anyway. Best of both worlds.
If I had to redo everything I would probably buy a NUC/mini-PC with a good CPU, 64GB RAM and low power consumption, stash a single huge SSD in there, migrate my VMs there and call it a day. But this is not a cheap setup.
