My feature rich Arduino web server at http://www.2wg.co.nz has been running for 18 days without error and runs with about 1K of free RAM.
Can you determine if the Arduino has stopped or just the Ethernet that has stopped?
If it is the former you may have to find a memory leak - start by checking free heap memory - if it grows until your system crashes you have confirmation of a memory leak.
If it is just Ethernet stopping you need to extract and observe your socket statuses.
The source code for my application is available to download on its website. You will find code to both extract memory usage and check Ethernet socket statuses in there.