What are good ways to statistically test for noise? I have read that highly autocorrelated data (such as stock markets) can be problematic when creating forecasting models essentially because we may infer that there is some signal or underlying model when in fact we are making sense of noise and the system is just highly autocorrelated.
I am looking to model sunlight irradiance which is highly seasonal on a yearly and daily level. Is there a good test to show to what extent noise is inherent within the system so that I can understand what the feasible limit for forecasting would be? It may very well be a silly question.

This is one of the points of time series analysis. The third edition of "Forecasting: Principles and Practice" by Hyndman and Athanasopoulos, freely available on-line, is a superb introduction to practical analysis of time series. Chapters 5 and 10 are perhaps most directly related to your question, showing how to generate plots showing prediction intervals into the future based on your model of the time series. This page, a bit old now, has many other suggestions for study.
