Can I square faces and edges of boards with only a #4 hand held plane? I've read a #4 can be used in combination with a #7, but a #7 is seeming out of the budget for me.
It is very much not ideal to do so, but it is always possible to flatten surfaces with planes shorter than the wood you're working on (obvious really, but many people don't see this and assume that long planes are necessary when in fact they're merely desirable).
To offset the physical limitations of a short plane doing this kind of thing you must adopt a careful, stage-by-stage approach to the work, stopping planing frequently to check progress. This is actually a lot harder to do than it sounds! But it's a must, and the same basic thing is the secret to the success of many hand-tool operations — doing the work slowly and methodically, checking at each break for square, flatness etc. and adjusting as needed for the next round of work. Every stage gets you closer and closer to straight or flat.
Other than that the main practical thing using a short plane is to identify and mark high spots and plane those areas only, where with a longer plane it would naturally bridge gaps and plane off those high spots automatically. This is illustrated in many guides to planing by a version of a simple drawing, here's one example from an old Stanley publication*:
Note it says easier, not only possible.
Paul Sellers would be a good person to focus on if you're serious about doing everything with a no. 4 (not just for this by any means but for this especially). He is a big proponent of doing a lot of work with the no. 4 where others would advise doing the least amount of work possible with one, using other planes as needed for each of their specialist roles.
Now as I read someone put it once, Paul Sellers can do jointing and flatten boards with a no. 4 because he's a ninja with a plane, and you aren't so you can't. While that's amusing and has an element of truth if you work as advised above you can achieve similar results I promise you.... but much more slowly obviously.
*There is a surprisingly amount of cornerstone advice on planing on just that one page so I'll link to the whole image it's taken from, here on LostArtPress so you can see the whole thing.
I couldn't tell from the original question whether you had a #4 already, or whether you're looking for a first plane to get started, say. It sounds like you've made the decision to dimension by hand, so that's a given and it wouldn't be relevant to say you could consider power tools that produce the needed result in more of a production manner and spend time saved toward finished product. Or that a jointer might not be as handy in a shooting board, for example (other purposes weighing in the decision).
I sure get a lot of enjoyment out of hand planing (just in its own right, he whispers). If you're just starting you'll be figuring out some of the things that are as important as the length of the plane (how to keep the blade sharp, tool configuration, the feel for planing technique, reading grain, how to produce parallel faces and edges once you've got a reference, etc.).
I've been doing a lot of planing but still consider myself in the seasoned beginner class :) You mention board in your post and I was curious what lengths you'd be working with. I feel comfortable dimensioning a three foot board with a "short" plane, say, but would run in terror from eight! I'm better at surfacing edges rather than faces and handling twist/cup puts my brain and patience into overdrive, even with an augmented tool set, meaning winding sticks.
There are some good articles suggesting a progression of hand plane purchases. For example: https://www.wwgoa.com/article/hand-plane-starter-kit-four-planes-for-hybrid-woodworking/ . I bought a 5 1/2 as my second plane and love it.