Can spurious correlations exist in the (theoretical) population? Is it possible that spurious correlations exist between two random variables X and Y in the (theoretical) population (here I mean purely by chance, not due to missing confounders in a model, etc.)?

With spurious correlation I assume that you mean the description on the tag

Nonzero correlation between variables $$ and $$ where neither $$ causes $$ nor $$ causes $$. May result from both variables having a common cause or the correlation being conditional upon a common effect of both $$ and $$.

The term 'spurious correlation' dates back to at least Pearson 1897 "On a form of spurious correlation which may arise when indices are used in the measurement of organs" where he describes the correlation between two variables that are a function of other uncorrelated variables, e.g. x/z and y/z will often be correlated even though x,y,z can be uncorrelated (and this has consequences to comparisons like, for instance, made in biology where relative lengths are compared)
Confounding
Richard hardy names the typical example of ice-cream sales and beach drownings. There is no causal relationship between the two, but because both have a causal relationship with a same third variable (hot weather) they will easily be found correlated. The spurious part here is that people might consider a causal relationship between the two. Correlation does not imply causation.
However, in this case the correlation is not wrong/spurious. Ice-cream sales and beach drownings are correlated. It is the interpretation that is wrong. This case would be better described as 'spurious relationship'.
Collider bias
A typical example of collider bias is the relationship between pretty people being often stupid and smart people often being ugly. The image below shows how this correlation might not be true and only seems to be present when we condition on a third variable.
For instance in the image below the green points are cases when prettiness + smartness is above some level. This might relate to the people in the media often being successful or talented people which requires a minimum level of a combination of certain traits. The correlation is only present in this sub-selection and underlying the relationship there is no direct causal relationship between prettiness and smartness. But even stronger, there is also no direct correlation between prettiness and smartness. It is the selection/measurement/presentation of the variables that makes the two correlated.

Note that collider bias is the opposite of confounding.

*

*Confounding bias: x and y are found to be correlated because both are caused by z.

*Collider bias: x and y are found to be correlated because they both cause
z and there is a selection effect based on z.

So confounding is an example of spurious correlation that may occur without a confounder variable.
Mis-specification of models
A way how correlations may be likely found, without an underlying direct causal relationship, but just statistical variation, is when correlation between time series are compared.
If we test the significance of a correlation like an observed Pearson correlation coefficient, then often the test assumes independent distribution of the pairs of data points. But, that assumption is wrong for time series. Time series often have some autocorrelation and the data points are not independent. Because of this, correlations between time series can have seemingly high statistical significance. You might call that spurious correlation.
See also this question Why do these time series appear to be dependent? and a classical work dealing with these type of correlations is written by Yule in 1926 "Why do we Sometimes get Nonsense-Correlations between Time-Series?--A Study in Sampling and the Nature of Time-Series". Or even earlier "The Elimination of Spurious Correlation due to position in Time or Space" by Student (WS Gosset) in 1914.
In the other cases there is still in some way or another some causal relationship present (only not directly between the two correlated variables). In this case with time series there is a complete absence of an underlying causal relationship, even indirect, but the correlations arise by chance (and the chance/probability is underestimated, it occurs much more often than what people assume with their test).
