There seems to be three mainstream opinions in the Rishonim and Acharonim.
Only if you're not planning on cheating someone
The Sefer HaChinuch (Mitzvah 258) says explicitly that one may not rely on such estimations by Choshen Mishpat. In his words:
ותזכר...שלא אמרו זכרונם לברכה על הכיוון כי אם בקירוב, ולכן אל תסמוך בזה בחלוקת הדברים בין בני אדם. ואל תתמה איך יכתבו דבר בלתי מכוון...ובזה מה שלא כיוונו בו מביא אותו לידי חומרא ואינו מזיק לכל אדם בממנו.
And remember...that [the Chachamim], may their memory be for blessing, did not say [these previously stated mathematical principles] precisely, but were only estimating, and therefore, do not rely on this for separating things between people. And do not wonder, "How could they write something imprecise?" ... And on this that which they were imprecise would lead someone to act stringently and would not harm a person with his money.
The Shulchan Aruch (CM 231:16) and Rambam (Geneivah 8:1) say the same thing, that people are very careful with their money, and they don't want to part with it that quickly. They are quoting a Gemara in BM 91a; if anyone wants to argue with this opinion, they have to be more stringent. And indeed, they are.
Only if you're dealing with π
The Maggid Mishnah in Hilchos Shabbos (17:26) says that one may rely on π=3, but he says in Hilchos Eiruvin (3:2) that one may not rely on √2=1.4. What's the difference? Perhaps it's because the former has a passuk (Melachim Aleph 7:23; see Eiruvin 14a) while the latter doesn't.
The Sha'alos U'Teshuvos Tashbeitz (1:165) gives two explanations of these estimations: either they're Halachos L'Moshe MiSinai, or they're there to help explain the sugya better and one may not rely on them. The first side clearly holds like the Sefer HaChinuch, Shulchan Aruch, and Rambam, but perhaps the second side agrees with the Maggid Mishnah that one may not rely on √2=1.4 but can rely on π=3. Or maybe that side agrees with the even more stringent opinion below.
In the aforementioned sugya in Eiruvin 14a, the Gemara opens up with the cryptic line: "How do we know [that the circumference of a 1-tefach circle is 3 tefachim]?"
Um, how do we know? Take a tape measure! (Or at least take a string, wrap it around a circle, and compare it to a ruler, or three fists lined up next to each other.)
The Tosfos HaRosh explains: Since π≠3, how do we know we can rely on this halacha l'ma'aseh? The Aruch HaShulchan (OC 363:22, YD 30:13) explains: Since π=3, the diameter is less than 1. How do we know we can make the mavoi permissible to carry in anyway?
These seemingly similar explanations could be arguing about this very point. The Aruch HaShulchan, who emphasizes the permissibility of the mavoi, would seem to hold that one may not rely on these estimations whatsoever. The Tosfos HaRosh, who asks how we know we can rely on π=3 but is silent about √2=1.4, could hold like either of the above opinions.
R' Moshe Feinstein (Igros Moshe, YD 3:120:5) asks whether one is obligated to use a microscope and other scientific tools to make his tefillin as square as possible. R' Moshe paskened that as long as one's tefillin match the 1:1.4 ratio mentioned by the Gemara, it's okay. He quotes a Brisker who implies that ideally one should not rely on such estimations, but technically it's okay. R' Moshe concludes in astonishment of how such a gadol like the Brisker Rav could possibly have said such a thing.
As they are discussing √2=1.4, it seems clear that R' Moshe holds like the above Shulchan Aruch, and while me'ikar hadin the Brisker agrees, ideally he would like you to hold like the Aruch HaShulchan.
The longer TL; DR
- The Sefer HaChinuch, Shulchan Aruch, Rambam, one tzad in the Tashbeitz, R' Moshe Feinstein, technically the Brisker Rav, and maybe the Tosfos HaRosh all allow you to rely on π=3 and √2=1.4 as long as you're not dealing with Choshen Mishpat.
- The other tzad in the Tashbeitz and maybe the Tosfos HaRosh allow you to rely on π=3 by non-Choshen Mishpat but not √2=1.4 even by the other three areas of halacha.
- The Aruch HaShulchan and ideally the Brisker Rav prohibit relying on these estimations.