Yes, Akamai can serve stale content if the request to the origin times out or produces an error code. Here's a screen shot of the "Caching" and "Cache HTTP Error Responses" behaviors.

Note, however, that your content will need to be fairly popular to remain in cache. If it's not popular, then it may be evicted before you're able to repair your origin.
A better alternative is to implement a Site Failover ruleset which allows you to serve your page with alternate content from a separate origin, or static assets from Akamai's NetStorage. Here's a screenshot of a typical Match of a failed origin and the standard Fail Over behavior.

The "Action" field provides the following options, which can each be configured to your needs:
- Serve stale content
- Redirect to a different location
- Use alternate hostname in this property
- Use alternate hostname on provider network
- Serve alternate content from NetStorage