I'm a beginner in time series analysis so I'm uncertain whether or not this data is stationary according to the ACF plot; I can observe a fast decay but on the other hand, a lot of the lines exceed -0.4 or 0.4. How would you conclude this?
When using the ACF plot to diagnose stationarity, usually the rate of decay (into insignificant territory) in ACF values is of greater concern that the actual ACF values. If the ACFs decay slowly, that is usually a sign of non-stationarity.
Without context of your data, and due to your ACFs' wave-like behavior breaching the significance range and high lag 1 ACF, there is evidence of non-stationarity; but I'd advise using a statistical test to be sure.
There are several statistical tests for stationarity, including:
The Ljung-Box test for non-zero correlations at lags 1-20. The null hypothesis is that the series is non-stationary. Small p-values suggest that the series is stationary.
The Augmented Dickey–Fuller (ADF) t-statistic test: The null hypothesis is that the series is non-stationary. Small p-values suggest that the series is stationary.
Personally, I use the ADF test most frequently.