I'm trying to track down a reference for the following claim, found in Kaledin's lectures Methods in Noncommutative Algebraic Geometry:
As it turns out, an arbitrary scheme $X$ also appears already on this level [viz regarding a differential-graded algebra as a `noncommutative variety']: the derived category $\mathcal{D}(X)$ of quasicoherent sheaves on $X$ is equivalent to the derived category $\mathcal{D}(A)$ of a certain (non-canonical) DG algebra $A$. The rough slogan for this is that “every scheme is derived-affine”.
This seems to harken back to classic papers by Kontsevich, Kapranov/Ciocan-Fontanine, but I struggle to find a clear reference; I understand the equivalence of categories between derived affine schemes and commutative differential-graded rings, but this statement is slightly different, and more general. $A$ is an arbitrary DG algebra over field of characteristic zero and the original scheme may not be affine. The author admits he did not feel obligated to cite anything, which is fine since he provided these high quality lecture notes.
This is Corollary 3.1.8 in Bondal - Van Den Bergh paper. The slogan also first appears in that paper (just after the statement of the corollary).
P.S. Let’s leave judging whether the slogan is “good” to the experts! Indeed that slogan permeates the school of “noncommutative algebraic geometry” of Kontsevich et al for many years, maybe one can say it is a “raison d’etre” of this school.