Check out the link. The key is to use a diluted solution to better kill the root vs. full strength that just topkills the weeds.

Here is a link that might be useful: Vinegar herbicide

One of the best weed killers is not an herbicide but your hands. You will not be taking a chance on killing your trees or poisoning your family. You can direct to the plant without overspraying. You may have sore muscles but you should wear them with pride when you look at what you have accomplished.

There is no way in the world that I can clear any area larger than a few feet with only my hands and/or a weeding tool. I have over 2 1/2 acres with lots of invasives and I need to get rid of them. How am I supposed to do that with my hands?

Vinegar is a broad spectrum plant killer, it will kill the top growth of any plant the spray contacts. However, this will not kill the plant roots, hopefully by killing the top growth you will also kill the roots but that might take several applications.
Mulches can suppress "weed" growth also.

"How am I supposed to do that with my hands" The same way the people reading your question would know that you are doing 2 1/2 acres. Since different plants need different types of plant killers, what plants are you trying to get rid of?

You will probably not be able to kill all of the plants at one time so start in one place, identify the plants in that place. Then research what will kill those plants. That large of space mulches are out. Give some suggestion of topography of land. Is it level, hilly, or what? More information from you is needed to provide you information.

Many "weeds" are annuals and you can control those easily by keeping them from spreading seeds, regular mowing and/or deadheading. Those "weeds" that are perennials can be mulched out of existence as well as prevented by spreading by, again, preventing the spread of any seeds, but mostly do not obsess about the "weeds" many of which are wild flowers.

Many of which are not wildflowers in my yard.

Again, minna, there simply isn't a cheap and easy solution to some of the issues that you bring up. Here are some of the questions that come immediately to mind when reading this particular post:

Are there desirable plants in this area that you wish to save? Or can something be suggested that will get rid of everything? Are these 'invasives' woody, herbaceous, annual or a combination of all three? Are they grassy weeds or broadleaf or both?

Most of us understand that we would need to attack this situation with a combination of different methods: hand culling, torching, mowing, mulch, and chemicals. The chemical component could be vinegar (since you are wanting something cheap and home-made). Effective and long term weed control will never consist of just one method.

Commercial herbicides are very complex and one needs to know what kind of plants need to go, what kind are staying, what kind of soil, and many other factors. Herbicides are also among some of the most toxic pesticides we use (in terms of humans health). I avoid them 'almost' entirely.

To address your specific question:" How am I supposed to do that [pull weeds] with my hands?" I agree that two acres or more is a large territory. But who says you have to do it all at once? The obvious answer is, "You do it a little at a time."

Rhizo makes good points. However, it should be understood that herbicides in general are not among some of the most toxic pesticides that are used. They are in general less toxic to humans than most insecticides and many fungicides, and many herbicides approved for use rate in the low to very low toxicity categories. There are exceptions of course! As for the original post, whatever technique is used to control the weed growth will need to be coupled with a weed suppressing mulch for long term management, preferably something like coarse wood chips thick enough to reduce regrowth and reestablishment.

Here is a link that might be useful: U of F Extension

I am in total agreement with both rhizo 1 and jayk. Determine what weeds are the problems for you, perhaps using the access to your local extension service and their advice. If you have 2-1/2 acres to remediate, a plan is needed. You need to know the weeds, the organic measures that you can take as well as the chemical applications that will bring this into a controlable situation where you can perhaps use cheaper and "green" controls into play. I would figure that in 2-1/2 acres you will need man-made products for awhile to get 'er under control. I, too, use as few man-made products as possible. We gotta get a handle on contaminating the environment, but it can't be "cold turkey."
hortster