Whenever I try to create a subsection within a section, it creates a completely new section outside of the section that I want the subsection to be contained in.
I can create sections within sections by the Insert - Section .... Is that what you are trying to do?
Note that MS Word “Section” is not quite the the same as Writer “Section”, so if you save your file as a Word document (or it otherwise visits Word context), the section structure will be “flatted out”.
Please give us more detail. Your version of the LO suite, your operating system, the exact procedure you use when you try to accomplish your goal. Also, edit your original question and attach a sample file showing what you describe. The file contains more than what meets the eye, and can give us clues to what it is that gives you problems.
I’m using the latest version of LO. I read that I could make subsections inside sections. But whenever I try to do it, I can’t. I have to use tables as a workaround.
It is absolutely impossible to determine the version number from this assertion: packaging proceeds at different pace on different platforms and varies with distro maintainer reactivity. Answer @keme1Ă©s request: OS name, LO version and, most important, save format.
Visual aspect of section nesting is different from structural nesting. This becomes obvious if your sections have not the same number of columns. Only the Navigator Sections displays the effective nesting.
Usage of sections makes sense primarily only to change locally the number of columns, but in nesting context, the result may not be the one you expect.Explain what you want to achieve so that we can suggest the most appropriate solution. And, remember, Writeri s not Word. All tips and tricks will work only if you save .odt.
Read where? Adding a link is something that should become an automatic sub-conscious action for everyone who refers to just anything these days 
Since there was no reference to the source where you read, we can’t know what “do it” really means - and even what “section” really means to you: a visual? a “section” in another software terminology? Insert → Section? Which keystrokes, cursor position, selection, menu/toolbar commands fit into these two brief words “do it”?
7.5.3.2 (X86_64)
Windows 10 build
.odt
I’m just trying to create sections WITHIN a section. A section of 2 or more columns inside of an already existing section.
That’s creating a new section on a separate line. It’s not really nested in the section. It doesn’t conform to the boundaries of the section it’s supposed to be inside. I have a 2 column section… and want a 2 column section inside that 2 column section. Not a new 2 column section on a new line separating the section.
So now you are starting to explain your expectations (still not clear, but a step forward), and claiming that it is “not really nested in the section” (which is obviously wrong, but shows that you have some specific end goal, that would be great if you explained clearly) 
In Writer, the sections are implemented exactly as horizontal stripes filling the whole page (or cell, or frame) width; you need to account for that. So a nested section (which is definitely nested, allowing to be e.g. hidden when the parent section hides) will still visually “cut” (“sect”) the parent section. In other words, the parent section does not create a bounding box for nested sections (unlike frames, or table cells, that do define a bounding box for contained sections). This allows you e.g. to have a 1-column inner section inside a 2-column parent section.
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There are many subtleties with sections. @mikekaganski explained the general rule.
Perhaps you mean that you want a 2-column section inside the outer section column, which would create visually a 4-column area if you have enough text to overflow the outer section column.
This is not possible according to the rule: a section extends from left to right edges of the outer “container”. I.e. inserting a 2-column section in an outer 2-column section keeps the same visual aspect. You don’t get a 4-column area where you have your inner section.
There is however one exception to the rule. When your page style requests several columns, sections under this page style extend only from left to right column edges.
Consequently, if your text layout is essentially 2-column, don’t create a section. This is better done in the page style and is more efficient since you avoid to nest “container” objects which always puts some stress on Writer with a risk of instability. Now, you can have sections jailed inside the page column(s).
For even better advice, describe the general layout of your document and the reason why you think nested sections are the track to go.
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Then I suspect that section nesting is not the way to go. I created a quick mockup of a 2 column page with a 3 column frame inserted. Is this akin to the layout you are looking for?
FramedColumns.odt (20.6 KB)
Note that text within a frame is in a way “aside” of the main text flow, unlike section content. This may be noticeable with range operations, and with cross referencing. Also, a section can cross a page break while a frame must reside within a single page.