Alexa only listens when a wake word is uttered by the user, and then it usually times out and the session will eventually close. So if you want to record input from the user that is not possible.
However, if you want to record a period of time like a timer without listening that is possible through a workaround. Each request has a timestamp, so you can record when a user says "Start" and save the timestamp in a database and close the session. When the user says "Stop" you can get that timestamp from the request and then do the math to get the time difference. Your sample utterances for this could be:
Alexa, tell [your-skill-name-here] to start
Alexa, tell [your-skill-name-here] to stop.
So you could match a device id or user id to these timestamps in a database. I've seen this done with another skill "Shower Buddy."
https://devpost.com/software/shower-buddy
You could use your own database for this or use Dynamodb.