Is it ethical to obtain journal articles from places other than the publisher's official sites?

Yes, it is absolutely ethical to use these files during your research. Many publishers allow academic authors to upload a so-called "preprint" version of a paper to their own institutional websites or put them in repositories, and of course these can be used by others in research.

There's maybe just two minor things to be careful about:

1. You should try to verify that the preprint version is mostly up to date with the published version, comparing e.g. publication dates.
2. It is common practice to put the "official" publication source, i.e., the journal version, into your reference list. Nevertheless, if you can't make sure that the versions are equivalent regarding what you're citing them for, it may be necessary to state that you were using a preprint version.

In some cases, authors may put papers online even though it's technically a breach of copyright. But since you don't know the author's agreement with the publisher, you have no way of checking that, and in any case, it would be the author or institution that violates copyright, but not you.

I wanted to chip in on this one. Where I live and do research, we have intermittent access to literature. For instance, right now, and probably until April this year, we don't have. Nonetheless, to exist as a scientist, one must have access to literature. So, I do everything in my power to get papers even though it's not "ethical" from the point of view of publishers.

I do not feel guilty about this, because my research is supposed to be done for the people, not for myself. If I get a good result that leads to some new technology, everyone, including the publishers, will benefit from it. Besides, it is about survival in research. Without constant access to literature, I could just as well close the shop and buy a mop for my new career.

I mostly agree with silvado's answer but want to emphasize a couple of different things.

1. It is ethical to look at freely available pre/e-prints, if they are permitted by the publisher. They often are, and often are not, but it is not your responsibility to check. However, you should cite published versions, which means you should look at published versions to make sure they contain what you want and section/page/etc numbered references are correct. At most universities, you can get access to the university e-library from home by logging in.
2. Many libraries (including mine) monitor how often journals are electronically accessed through the library subscription, and use this to determine the value of this journal subscription. Journals with less perceived value (from the library's point of view) may get cut to account for rising costs of other publications and/or budget cuts. So it can be beneficial for both the journal and the university for you to make the effort to download the article through the library subscription.