I must check url statuses in scheduled job, but their amount is too big (exceeds limit, which is 100), so I tried to use Continuation object, but I'm not really sure if is it possible to use Continuation without Visualforce page? I want to check statuses on a weekly basis and store results in objects' fields. If there's another solution, please give me a hint, because I'm quite new to Salesforce.
If you are running into limits on the number of callouts you can make in a scheduled job, the solution to execute a batch job and set the batch size to 100 (or some smaller number if you make more than one callout per record):
Database.executeBatch(new MyBatchUpdateClass(), 100);
The second parameter controls the batch size. A Continuation will not help you work with callout limits. You'll need to implement the Database.Batchable interface in your class.