Is there any way to play AOE2 without a CD legally?
Buying it on Steam isn't a bad idea. There's a good community that sprang up around it, and you'll get Steam multiplayer along with some slightly better graphics. They also actively update AOE2:HD based on requests from players with things you won't get from the CD.
If you want to run what you have without a CD, I would recommend downloading the unofficial/soon-to-be-official expansion Age of Forgotten Empires It includes a color fix for Windows 7 and removes the need for a CD.
In most places it is legal to make copies of software for personal use so long as only one copy is run at one time. I am not a lawyer and this isn't legal advice, but I believe this is true in the United States, Canada, and Europe so long as the CD doesn't have any copy protection.
Therefore, for personal use, in such places you can legally make a binary image file of the disc (an "ISO") and then use virtual drive software to load the image so that it appears to be a CD drive with the disc in it. I've done this personally with my very played copy of Oblivion since I started to get concerned about disc damage.
I won't recommend any particular software because it depends so much on your exact OS and sometimes on you disc drive, but any CD cloner and any virtual CD drive software will do the job.
Back in the days when I played it with friends in my LAN I made an image from the CD and mounted it with daemon tools. Worked fine.