I am new to ARIMA, and I am trying to understand these lag plots. Are the following ACF and PACF suggesting that the lag of my time series is 4? If I am wrong, please help me understand these plots.
The threshold statistical significance of the autocorrelations has been noted in the comments and in another answer. What looks interesting is that the autocorrelations in Lag 4 and Lag 8 persist also in the Partial ACF.
Reality should come into play at this point: what are these data? By whatever knowledge you have on the process they measure, is it reasonable to expect that the current level should depend on Lags 4 and 8? If yes, the low estimated strength of autocorrelation is not necessarily an artifact, but an indication that said autocorrelation exists but it is low in strength.