Since I'm just me, my parents help me out a bit with my morgage (advance on my heritage), the plan is to try to get my morgage down to a more manageable amount until I fully take it over in a couple of years, when my parents both retire (by then I will have gotten a raise at my job). The housing market over here is near impossible if you don't have a partner to share the costs with and since both my sisters are settled down, they didn't want me still renting a spare bedroom.
Other than that I would say I'm pretty independent. I moved away from home at 16 and learned on the go how to take care of myself. I love my family dearly and I'm really close to them, especially my mom, but loving and depending on people are two different things. I am very open about me and my life, but I've learned that I'm also very capable of handling the things thrown at me. If people can and want to help, I won't deny them the chance to do so, but at the same time if no one wants to or can help, I manage to find my own way.
I don't believe independent means never accepting a hug, or never letting someone help you move a couple of boxes. I believe independent means being capable to do what you need to regardless what support you have or don't have. Last year I had to move to my new place, no one was able to help me out (my family was on the other side of the country to celebrate my nephew's birthday, and my friends were either pregnant or away for the summer), so I had to organize, pay for and do the work myself. And I made it work. That is what independent means to me. Not crumbling under the pressure when I have no one that can help me.