I have heard exactly one person systematically, non-ironically and non-mockingly pronounce dass as /da:s/, i.e. with a long /a:/. However, that person also systematically pronounced the /a/ sounds in ein nasses Glas identically: as /a:/. Thus, this person is not a good source aside from giving anecdotical evidence.
Every single other person I spoke to pronounced dass with a short /a/ systematically. Now the difference between short and long /a/ is not always phonemic in every dialect and accent of German, but there have been people who distinguish /a:/ and /a/ and those that don’t among those I listened to. Thus, I suspect that you have been taught incorrect German to better rationalise the pre-1996 spelling of dass at school.
There is no etymological reason for the spellings of das and dass to differ; the Grimm dictionary notes:
DASZ, conj. gebildet aus dem neutr. des pronom. der in seiner relativen bedeutung, wie im griech. ὅτι, lat. quod, franz. que, engl. that. im goth. wird das relative bedeutung wirkende suffix ei angehängt, þatei, in Muspilli daʒî 12. im ahd. und mhd. gilt daz für die conj. wie für das pronomen: im nhd. hat man eine unorganische, für die aussprache gleichgültige unterscheidung eingeführt, indem man die conj. dasz, das pron. das schreibt.
The two stem from the same root and differentiating the spelling is solely to help determine whether the word in question is a conjunction or a pronoun (including a relative pronoun). Since they derived from the same stem, there is no reason to pronounce them differently.
The question does remain why daß was the correct spelling pre-1996 and dass post-1996. The former spelling rules used the Adelung s-spelling while the ones currently in effect use the Heyse s-spelling. They differ in the question whether ß or ss should be used after short vowel sounds at the end of a morpheme. The Adelung spelling — which requires ß at the end of morphemes strictly — has its roots in blackletter typing, which coincidentally is also the root of the ß ligature. Blackletter, which also distinguished between ſ and s, had its own reasons for following what would later become the Adelung rules.
Since a distinction was sought between the conjunction dass and the pronoun das, it only made sense to use a different depiction of the s sound — and blackletter and Adelung only allowed ß as a variant. Hence there was no discussion, the length of the vowel remained obfuscated and it was probably systematically pronounced with a short a. In the 1996 reform, it was decided to keep the distinction between conjunction and pronoun — not only had it already been established in German writing but it also proves to be a slight aid to the reader. However, the pronunciation of the a in dass was obviously short so the only way to keep up the distinction was to turn daß into dass.
: And then there are those that mockingly and ironically pronounce /da:s/ if they are reading texts in the pre-reform spelling and come across daß – much like they will pronounce muß as /mu:s/ in the same context. That this joke works is a strong indication for the correct pronunciation to be with a short /a/.
: But even if they were two different words from two different stems, that does not mean that their pronunciation cannot merge to give a homophone. German has a few homophone pairs that derive from different stems, e.g., Reis (-korn) and Reiß (-leine).