There are actually lots of ways to power things, and we can take effectively a short trip through history to find lots of them.
Flowing water exerts a force on anything that opposes the flow - and this was in fact one of the most widely valued and used sources of power in human history, found in everything from boats that follow currents to water wheels, irrigation, aqueducts, etc. Water wheels were one of the most useful early power sources to power things like mills on large scales. The biggest downside is mother nature is not terribly obliging about where she puts flowing water, so you don't get any real mobility options. If you want something worked on you have to drag it to the power source.
Water screws may also be of interest to you, but they are a better example of a pump than a power source, I suppose. You can hook it up a bit differently to create a spinning turbine with the power of water, but that's not that different than a water wheel at that point.
The stirling engine is one of my favorites, and it's still finding various uses today, partly because it just needs a heat differential. If you can get one piece hot and keep one piece less hot - the bigger the difference, generally, the better - you can keep the engine going. You are in a constant fight with thermodynamics, though, so it can be pretty tricky to keep things running. Recent inventions have actually used this as a positive side effect, in that they use water as the coolant - with the side effect of generating clean water as well as power from any form of heat source available (dung, straw, wood, coal, whatever).
Any set of chemicals that produces a thermal or expansion reaction can be used as a power source. Mix together acid and something for it to react to (like marble chips) and you can get pressurized gas (like CO2) or heat you can use to power whatever you like. Anyone can do this, but only someone with a good knowledge of chemistry could select specific mixtures of chemicals to use to produce the right combination of heat/pressure/timing to be very useful.
The downsides are primarily availability of materials (like hydrocarbons or strong acids) and side-effects (like corrosion, accidental explosions, toxicity, etc).
Does Home Depot carry bed sheets? Sail boats are ancient for a reason - the wind is a great source of power for a boat. Beyond boats comes windmills, which again are ancient for the reason that they are really such a simple concept, yet if you are in an area with strong winds it is an undeniably useful power source. The downsides, naturally, are complete reliance on the environment to give you access to steady, strong winds. Great in some areas, useless in others.
Generating electricity from sunlight would be a bit unreasonable for a simple DIY machine power project, but solar ovens and the like can be a simple source of heat to use in heat engines, or just to build a simple oven. Hard to realistically power a tank with solar power, but it does have it's uses.
It's like hand-powered, but by a non-human! Humanity has a long history of using animal species like oxen, horse, donkeys, etc to provide mechanical power far beyond what individual humans can provide. You'll need a good organic power source (food) and animals are famous for being dangerous, but you can certainly build your own yoke from simple materials. Power a mill, pull a vehicle, whatever you like!
Inventions like the trebuchet rely on counterweight - ultimately, gravity - to power some insanely powerful machines. You can use any other source of motion to 'store' energy in the form of something heavy lifted up high - or a pulled string/spring - to power a sudden, controlled release of energy. The sky is the limit, so to speak.
The downside is that the power tends to be short-lived, and only deliverable in a single concentrated dose, with considerable wind-up time in-between uses.
Tom Sawyer Power
When all else fails, convince someone else to crank something for you. You can be clever, like Tom Sawyer, or a psychotic slave driver and force someone else to do it. This is really just another form of hand-power or animal power, but "Tom Sawyer Power" just sounded like a fun category.