Perfect complementary sequence sets (PCSSs) are widely used in multi-carrier code-division multiple-access (MC-CDMA) communication systems. However, the set size of a PCSS is upper bounded by the number of row sequences of each two-dimensional matrix in the PCSS. Then quasi-complementary sequence sets (QCSSs) were proposed to support more users in MC-CDMA communications. For practical applications, it is desirable to construct an $(M,K,N,\vartheta_{\max})$-QCSS with $M$ as large as possible and $\vartheta_{max}$ as small as possible, where $M$ is the number of matrices with $K$ rows and $N$ columns in the set and $\vartheta_{\max}$ denotes its periodic tolerance. There exists a tradeoff among these parameters. Constructing QCSSs achieving or nearly achieving the known correlation lower bound has been an interesting research topic. Up to now, only a few constructions of asymptotically optimal or near-optimal periodic QCSSs have been reported in the literature. In this paper, based on polynomials over finite fields and Gaussian sums, we construct five new families of asymptotically optimal or near-optimal periodic QCSSs with large set sizes and low periodic tolerances. These families of QCSSs have set size $Θ(K^2)$ or $Θ(K^3)$ and flock size $K$. To the best of our knowledge, only a small amount of known families of periodic QCSSs with set size $Θ(K^2)$ have been constructed and most of other known periodic QCSSs have set sizes much smaller than $Θ(K^2)$. Our new constructed periodic QCSSs with set size $Θ(K^2)$ and flock size $K$ have the best parameters among all known ones. They have larger set sizes or lower periodic tolerances. The periodic QCSSs with set size $Θ(K^3)$ and flock size $K$ constructed in this paper have the largest set size among all known families of asymptotically optimal or near-optimal periodic QCSSs.