How to Find 24 Hour Daycare or Overnights?

Are you talking about kenneling your child? I mean like drop Fido off here, drop junior off here and go on vacation by yourself?

That does not exist.

The best you could hope for is cultivating an adult babysitter that would be willing to take the child for an extended period. They will charge quite a bit for such services.

When our kids were younger we just booked an extra room and took a sitter with us. I would imagine that is cheaper and safer.

If you don't have family in the area who would be able to help you in that situation, I would find a reliable, older babysitter you can trust. I would search in your church (if you go), or anywhere where you are actually in contact with people. You might try one of the baby sitter websites. You will need to use her (or him, if you find a guy who fits the bill...) a LOT so that your children come to think of her as practically family. I wouldn't feel comfortable leaving my kids with anyone less for that long. If you maybe have a friend with a kid your child's age, maybe you could even 'swap' babysitting time, so that when they get left behind when you go (not that that is always a bad thing... parents do need time away after all...) they can have fun too.

Following Jenna M's guidelines, if you don't have family or friends who can do it, I would start looking at in home child care places. You will want to take the child there regularly during normal business hours for months before going on the vacation to build a relationship of trust. All of the people I know who have gone on vacation without children left their children with family. How old is the child?

Yikes, I couldn't do that, how scary for a child

Wow, shocked at some responses myself but then again perception is everything. As a military family with family nowhere close we have a 24 hr provider who we found on the internet. They go to her on a regular basis and when me and my husband are both out on travel they stay with her if need be. We do have friends who help out but they have their own kids, adding my little ones would be hard for more than a day at a time plus I want my kids to keep their routine as much as possible when we are gone. The charge is usually something that is negotiated ahead of time and reasonable because we have built a really good relationship with her. I am a firm believer that your care providers are an extension of your family when you have no family around.

depends on the child's age. First: Family/friends. If he/she is older maybe offer to pay the parents of his close friend to spend the week while you are out of town. Each family is different, so use your intuition. If he is really young, could you take him with you and get a babysitting service from the hotel?

I'm really not sure that actually exists. I don't think there are many people who would leave their child with strangers for 24 hours or more...

Yes it does exist. There are several 24 hour care in-home centers here. They cater to the families that work the nightshift/odd hours like nurses and military.
My in-home provider has done this for a few families.

Start with care.com.