You haven't hit the Salesforce API limit, which is generally quite high. You've hit the limit for the Chatter REST API, also known as the Connect API. This API has a per user, per application, per hour request limit:
Chatter REST API requests are subject to rate limiting. Chatter REST API has a different rate limit than other Salesforce APIs. Chatter REST API has a per user, per application, per hour rate limit. When you exceed the rate limit, all Chatter REST API resources return a 503 Service Unavailable error code.
The linked document has a number of recommendations for avoiding this rate limit:
- If you hit limits when running tests, use multiple users to simulate a real-world scenario.
- When polling for feed updates, do not exceed one poll per minute (60 polls per hour). To return more results in 1 request, increase the page size.
- When polling for private messages, do not exceed 60 polls per hour.
- To avoid making multiple requests, cache metered static assets such as file and dashboard renditions (group and user profile pictures are not metered).
- Each developer on a team should set up 2 connected apps: one for automated testing and one for manual testing and development. Don’t share connected apps with other developers.
- Use a unique connected app for the production environment.
- Don’t share connected apps between applications.
Review the list of Chatter REST API resources to determine what you're calling that is subject to these enhanced limits.