You can build an ar for around 400$ pretty easily
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With all due respect, that is an incredibly misleading statement. Sure, you "can" probably piece together a rifle that cheap using all low end stuff like PTAC, but there is no way to build a complete rifle including sights and magazines for that. Just a set of budget sights are close to $100 by themselves, same with an optic. The lowest I would ever tell someone they can budget for a decent AR is around $600 for something like a M&P Sport, or a similar build. Very few first time people would be able to build one cheaper without having the experience of knowing what to buy, and where to source the best deals for quality components. For instance, it's very tough to even find a decent complete upper for that $400 you quoted, because just a decent BCG is going to be about $100 minimum. If I was going to build more than just a range plinker, not something to defend my life with, it wouldn't be with extreme budget components, that's for sure!
OP, as stated by others, about the only decent center fire rifle that meets your criteria is going to be something like the Hipoint carbine. The nice thing about that is if you buy it in the same caliber as whatever handgun you plan to buy (or already own), you don't have to stock too many calibers. A pistol caliber carbine is also practical for home defence because you have more accuracy and control than a handgun, without such a powerful round that it is going to keep going and kill one of your neighbors like a 308, 5.56, etc. can. As mentioned, a decent pump shotgun can fit your budget and makes a great HD tool as well as a fun range toy. If you really want an AR, then you should plan on a minimum of about $1000 to have a decent rifle with appropriate accessories to make it serviceable as a HD gun and good for range work too.
A few guys mentioned the Sub 2000. That should be considered as well because it has the advantage of being able to use the same magazines as several popular pistols including Glocks, Beretta, S&W, etc. depending on the version, pre-ban high capacity mags are also available, which you can't have with the Highpoint. The major downside is that the Sub 2000 is not easy to find, and even though msrp is in the $350-$400 range, it is nearly impossible to actually buy one, even used for that price because of the high demand/low availability.