Is it possible to plug 10 cat5 (network) cables that come from 10
different DSL modems to an HP server so I have 10 different broadband
lines connected to one single server?
Yes, it's a terrible idea but yes you can.
If so how?
Buy some multi-port NICs
Any suggestions from geniuses in Networking?
A hint of sarcasm there but I'll rise above it - yes, here's a suggestion, don't do this like this at all, it's silly. Why do you think you need 9 x DSL lines? If this is for remote access are you expecting all bandwidth to be used on all lines all the time? Even if you really do need (or think you need) these nine lines why not connect them all to a single switch and connect the server to the switch via a single/dual normal 1Gbps NIC? If you need to segment them then just put each port into its own VLAN and trunk the uplink with .1q?
Of course this provides no resilience, you don't mention how important that would be.
In terms of server spec then sure HP's are great (the DL380 Gen9 is the most popular server in the world as it does 99% of what 99% of people need done) and although you don't mention which hyper-visor you're planning on using I'd imagine 256GB will be sufficient. Be aware though that HP doesn't support non-HP disks of any type, and we strongly advise you to not shoehorn your own disks into HP disk caddies - otherwise one day you'll be back here asking for data recovery and/or performance issue assistance (search the site for dozens of requests like this). But do make sure you use RAID of some form, and again we recommend here not to use RAID 5, its time has passed for dull technical reasons - try to stick to RAID 1/10 or 6/60.
Oh and one final thing - when you sign up to this site we put a bunch of words in front of your face, most seem to ignore them but some of those words state that this site is very specifically not for 'noobs' or beginners. You might wish to re-read that text.