I think some of the responses you are getting here are ridiculously harsh, rude, and just a bit silly.
I think that it is a very useful skill and valuable tool for happiness to be able to entertain yourself, and at 10/11 it should be possible - though some children are certainly much better at it than others! I don't think you should have to take children out every day, provided you've got a decent amount of space at home and in your garden.
We've recently cut back a lot on formal childcare and take turns to work at home while our children 'potter'. We're around, and spend some time with them, but we expect them to largely entertain themselves.
I STRONGLY recommend arranging as many meet-ups with friends as possible. Obviously if they can go to another friend's house that's easiest, but I actually find my children are rarely any bother at all if they have a friend over here - that certainly isn't always true if they are on their own! I have a 10 yr old boy and an 11 yr old girl and find it works best if one goes to a friend's house while the other has a friend over, as any problems come from clashes between the two!
They are also now old enough to walk to the local shops, park, library etc by themselves - with clear deadlines and a phone. I've just sent my 11 yr old DD to the shops to buy some biscuits as we have friends coming round for a cup of tea later.
You've got a 2 yr old so you've got it much harder, but you can totally ignore those voices giving you are hard time for not wanting to entertain a mithering 10 yr old the whole time!