TL;DR: measure your 'Heart Rate Variability' with a smartphone app or wearable device
The longer answer is more complicated. This meta analysis suggests:
At present, there is no accepted standard for stress evaluation.
Heart Rate Variability
This study concluded that:
the current neurobiological evidence suggests that HRV is impacted by stress and supports its use for the objective assessment of psychological health and stress.
So Heart Rate Variability could be a useful biometric indicator. However, this study says that many things can cause high HRV:
stress, poor sleep, unhealthy diet, dysfunctional relationships, isolation or solitude, and lack of exercise
so it's possible your HRV could go up independent of your stress levels (so use with caution).
Harvard Health suggests how to measure HRV:
The gold standard is to analyze a long strip of an electrocardiogram done in the doctor’s office. But in recent years, companies have launched apps and wearable heart rate monitors that do something similar.
Other biometric indicators
The University College London measured stress levels in experiment participants by using:
- Pupal dilation
so these may be possibilities too.
Perceived Stress Scale
There is an old fashioned way... the Perceived Stress Scale was:
published in 1983, and has become one of the most widely used psychological instruments for measuring nonspecific perceived stress. It has been used in studies assessing the stressfulness of situations, the effectiveness of stress-reducing interventions, and the extent to which there are associations between psychological stress and psychiatric and physical disorders.
Here's a copy of the actual test. The result can be interpreted with detailed scoring:
or, using these rules of thumb:
► Scores ranging from 0-13 would be considered low stress.
► Scores ranging from 14-26 would be considered moderate stress.
► Scores ranging from 27-40 would be considered high perceived stress.
It looks like a test you would take approximately monthly.