The scriptural foundation for this idea appears in Yechezkel 11:16, where God declares that even after scattering Israel among the nations He has become for them a "diminished sanctity" (מִקְדָּשׁ מְעַט) in the lands where they have gone — a verse the tradition reads as God Himself becoming the sanctuary for His people in exile.
The Gemara in Berakhot 8a draws out the post-churban implication directly: since the day the Temple was destroyed, the Holy One has in His world only the four cubits of halacha — so that immersion in Torah becomes the vessel through which the divine presence continues to rest in this world.
Nefesh HaChayim, Gate I 4 builds the inner logic of this equation: when a person sanctifies himself through the fulfillment of all the commandments, he himself becomes the actual Holy Temple, since the verse says "ושכנתי בתוכם — I will dwell within them," not within it, meaning the dwelling of the Shekhinah is within each individual person.
Tanya, Part I; Likkutei Amarim 36 frames the entire purpose of creation as God's desire for a dwelling-place in the lower worlds, and Tanya, Part I; Likkutei Amarim 34:4 specifies the post-churban reality: since the Temple was destroyed, God has no dwelling and place of unity in His world except through Torah — so that each person who occupies himself with halacha becomes that dwelling.
מכתב מאליהו, חלק א' עמוד רכח, וראה שם עוד makes the inner-outer relationship explicit, citing the Nefesh HaChayim: the Mikdash built in Jerusalem was only the external vessel for the internal sanctuary in the human heart, and when Israel corrupted the inner sanctuary the outer one could no longer stand — leading to the reverse conclusion that rebuilding the external Beit HaMikdash depends entirely on rebuilding the inner temple of the heart.
מכתב מאליהו, חלק ה עמוד צד develops this further, explaining that the heart itself is the Mikdash shel ma'alah — the heavenly Temple — with a gate that remains sealed throughout the week and opens only on Shabbat and Rosh Chodesh to admit the revelation of God's glory into the person's inner being.
רבינו ירוחם, דעת תורה דברים עמוד רד frames the entire enterprise of the Mishkan's construction as the work of cleansing and purifying one inner corner from the stain of sin and restoring it to its original holiness, so that it becomes fit for the Shekhinah to rest there — making all of a person's spiritual labor an act of creating a place that is mikdash, through which divine blessing can flow.
Yeshayahu 57:15 supplies the theological ground: God who dwells on high and in holiness also dwells with the contrite and the lowly in spirit, reviving their hearts — so that the broken and humble heart becomes the very site of divine habitation, the inner sanctuary rebuilt precisely through its brokenness.
Rav Ephraim Greenblatt (רבבות אפרים ח"ב סי' ו' ד"ה ויש) addresses questions of netilat yadayim and the maintenance of personal purity in various circumstances, which bears on the practical dimension of preparing the self as a vessel of sanctity in the absence of the Temple service.