“It's been one thing after another all day.”

You've had a busy day at work, with lots of different people asking you for help with different problems. Now it's the afternoon, and you're talking with a coworker. You want to tell her about your busy day. You say:

It's been one thing after another all day.

It's been (adjective/noun)

You use "has been" to describe a situation that started in the past and continued until now. The situation can be something that happens continuously:

They've been arguing with each other.

She's only been gone one day.

Or it can be an action that is repeated over and over for a length of time:

I've been going to that grocery store for over 25 years.

 The phrase "It's been" is used to describe your general recent situation. It's like saying "my life has been ___":

It's been tough looking for a job in this economy.

It's been a fun vacation, but honestly I'm ready to get back to my routine.

one thing after another

"One thing after another" means a series of small problems or annoyances. You use it when problems keep happening again and again.