First, let's make sure you have an eMac. Please don't think I'm being condescending; rather I've been here a long time and have seen the confusion Apple has caused with its similar-sounding product names. Some people interchange "eMac" and "iMac" especially those whose native language does not use the "i" sounds as in "site." This is the eMac, all 50 pounds of it:
I still own 100 pounds of working eMacs!
Before embarking on this project, we should precisely identify the eMac you have. There were four eMac variants and two had serious (and project-ending) hardware issues. One of those with issues had a fatal flaw that is today not practically repairable. Please use this archived Apple support article to identify your eMac so we can advise if this is worth pursuing:
eMac: How to locate the serial number and identify your model - Apple Support
Let us know what model that shows but please do not post your serial number.
Can I use macOS Server to do
this? If so, which version since I am
running Snow Leopard or Lion?
If you have Snow Leopard or Lion on the eMac, it's not an eMac--one reason for my initial query above.
Because the eMac is one of the last PowerPC Macs, it can run no OS higher than 10.5.8 "Leopard" due to hardware constraints, even on a late-model unit. Early eMacs with 700 and 800mHz processors can run no higher that OS 10.4.11 "Tiger." Snow Leopard and Lion require an Intel processor, something an eMac never had. So you will be looking for of 10.4 Tiger Server or OS 10.5 Leopard Server depending on which eMac you have. Legitimate copies of that package are selling for about 10X the value of any eMac today:
https://www.amazon.com/s/ref=sr_nr_n_0?fst=as%3Aoff&rh=n%3A229534%2Cn%3A229660%2 Ck%3A10.5+server&keywords=10.5+server&ie…
Also,
should I upgrade my modem? I still have
the original installed (56k v.92 modem).
eMacs had an Ethernet port. Even though they are the slower 10/100Base-T protocol, I would used that for any networking.
eMac modems are slotted and theoretically replaceable but there are reasons to go in a different direction:
1) the internal modem is a proprietary design and you can't get one any faster than what you have. An external USB modem might work if yours has USB 2.0 ports but I doubt you can get much faster than the v92 you have now.
2) accessing the modem requires a fairly involved takedown of a computer that is notoriously difficult to reassemble without breaking something you really need, like the power button.