Probably the most common view today – boosted by James M. Hamilton’s 2006 book God’s Indwelling Presence – is no.
John de Hogg (lecturer in OT and Hebrew at RTC Australia) sums it up:
‘Michael Green in his book I Believe in the Holy Spirit presents a rather common view. “On the whole you had to be someone rather special in Old Testament days to have the Spirit of God. A prophet, a national leader, a king, perhaps some specially wise man (Prov 1:23) or artistic person (Exod 31:3) – in which case you would be beautifying the Lord’s Tent of Meeting or enunciating the Lord’s wisdom. But the Spirit of God was not for every Tom, Dick and Harry… The gift of God’s Spirit was on the whole to special people for special tasks. It was not generally available, nor was it necessarily permanent.” This common view says that OT believers were not indwelt by the Holy Spirit and that OT believers did not experience the active, internal, personal ministry of the Holy Spirit. These blessings came as a result of Pentecost.’ – Vox Reformata, 2012
However, de Hogg then goes on to refute that position. While his article is fairly helpful (and interacts specifically with Hamilton), there’s an absolutely storming one by Walter Kaiser, also taking a contrary view:
“…How could all of these old covenant persons have believed and been enabled to live sanctified lives if the Spirit of God did not dwell in them? Must we say that an Old Testament believer was able to please God spiritually and to be sanctified in the presence of God without the indwelling presence of the Holy Spirit? Did not Enoch please God (Heb 11:5)? Was not Noah one who walked with God and who was righteous and blameless (Gen 6:9)? Did not Joseph resist temptation (Gen 39:9)? Was not Job one who turned away from evil, one who feared God and was blameless and upright ( Job 1:1)? Did not David pray, do not ‘take your Holy Spirit from me?’ (Ps 51:11). Did not the prophet Isaiah teach that the people ‘rebelled and grieved his Holy Spirit’ (Isa 63:10), the same Lord who had ‘set his Holy Spirit among them?’”(Isa 63:11). – Evangelical Quarterly 82.4 (2010)
Both articles deal with verses in John which would seem to be the strongest arguments against such a position.
Kaiser also deals very helpfully with the broader question of ‘What is new in the New Covenant?’, and specifically, what is new at Pentecost (including a very helpful quote from Thomas Goodwin, also cited in Smeaton’s classic work).
This 1976 letter by John Piper is also pretty helpful.