This is definitely possible! Do you already have the means to transport files from the internet to your Color Classic? If so, it is a matter of plugging an ethernet cable into your router, installing an FTP program such as NetPresenz on your Color Classic, and configuring a few network settings in a program called MacTCP. You would then be able to transfer files from your modern computers directly to your Color Classic over the network.
Netpresenz is a free file sharing program for your computer: http://www.stairways.com/main/netpresenz
I also recommend installing MacTCP 2.1, which is a patch for the MacTCP program. This software version makes configuring the network settings much easier. Here is a link to MacTCP 2.0.6 which is to be installed prior to updating to version 2.1.
https://macgui.com/downloads/?file_id=12993&keywords=mactcp
Here is a link to the MacTCP 2.1 patch
: http://www.mactcp.org.nz/mactcp.html
You might also want to take a look at this website which has lots of information on the topic of networking vintage Macs:
http://www.applefool.com/se30/#ftp
I went through a similar process a few months ago when networking my Macintosh SE/30. Some of the information on that forum thread may be of use to you too. Luckily, I had a much much easier time networking my Color Classic than my SE/30!
Configuring the system network preferences can be a little tricky, but it is certainly not impossible. I am more than happy to further elaborate on this if anything I said does not make sense, or if you have any questions!