Your question actually includes a serie of good question.
1- Can I use a modem designed for to go over the Plain Old Telephone Service (POTS), like a 56K modem, as an ADSL modem?
The answer to this is no. The ADSL signal and the voice signal (used by the POTS modem) are two different kind of signal and one cannot be used to transmit data over the other.
2- Can I use a linux box as an ADSL to modem bridge? (i.e. using a computer running Linux to bridge the gap between the telephone network and an internal network)
The answer to this is yes. To do it you'll need to equip the computer with an ADSL interface card. The Linux Kernel supports a number of them. Be aware that unless your Internet Service Provider (ISP) can provide you with an interface card and offer supports for Linux computer (I don't know of any that do) you'll be on your own to make it work.
3- Should I get into trying to get that working?
If you're looking for a solution with a minimum of maintenance, the answer is, in my opinion, no. It's simpler to have a dedicated router and dedicated ADSL modem. That configuration provides a point of demarcation if you have issue. Connect a computer directly to the ADSL modem, make the connection and see if you can reach the internet. If you can connect, you know the issue lies with what's behind the modem (either the router or the computer behind the router). If you cannot connect directly, then it's either the modem, the connection of the modem to the phone line or a problem at the ISP level.
A few reference on having a direct DSL connection to a Linux computer