Q:

What is the relationship between government expenditure and the quality of life?

Can we say, in general, what is the relationship between relative government expenditure (or the total taxation) and the average quality of life?

A:

Triggered by the somewhat arbitrary selection of countries in the OECD graph in Bobson's answer, I found that data are easily available for much more countries than just the OECD countries. 
Summary: there is some positive correlation between tax : GDP ratio and happiness score, but the correlation is too low to allow meaningful conclusions about happiness score from known tax : GDP ratio for a generic country. 

Putting together with Wikipedia List of countries by tax revenue to GDP ratio
the [World Happiness Report 2017 table also from Wikipedia]https://en.wikipedia.org/wiki/World_Happiness_Report#2017_report) (the 2017 report is on happiness data 2014-2106 which matches better than the 2018 report with the age of the tax revenue data)

I find:
 

blue: OECD countries (to ease comparison with @Bobson's answer)
red: countries where happiness (+) or tax revenue : GDP ratio (x) is missing

With the unaided eye, I'd say that there's a maybe lower line of very roughly 1.75 happiness points + 9 ⋅ total tax revenue/GDP. Very few countries are below that:

There may or may not be a similar upper limit as well at roughly 5.25 happiness points + 12  ⋅ total tax revenue/GDP (dark green line).
The five (dark green dots) countries above are the United Arab Emirates, Bahrain, Qatar, Kuwait and Saudi Arabia, i.e. the rich Arab oil countries that probably use lots of oil sales revenue instead of taxes. 
And, anyways, even the most happy countries so far do not manage to get a happiness score of more than 7.6 (red line). 
If I do a linear regression of happiness score over Tax revenue : GDP ratio for all countries where both happiness and tax revenue are available, I get the following picture:
 
Regression formula is 4.425 happiness score + 4.36 ⋅ total tax revenue/GDP. Both intercept and slope are highly significant*, and the corresponding 95 % confidence interval for the regression line is the reddish shaded area in the graph. However, knowing that on average over almost 150 countries happiness correlates with tax revenue to GDP ratio (correlation coefficient is, btw, 0.49) doesn't help us saying anything about a single country. For that, we'd need to look at the prediction intervals (blue shaded area). 
That tells us that for a generic country with 25 % of GDP going into taxes, we'd expect a happiness score somewhere between 3.5 and 7.4 (95 % prediction interval). Happiness score of 3.5 is what Guinea, Liberia and Togo in West Africa have (Syria is 3.46). 7.4 basically scatches the very top we actually have anywhere worldwide, it is between the Netherlands and Finland (topmost was Norway with 7.54). In other words, we'd expect it to avoid being one of the 6 most miserable countries, and also not to make it into the top 5 countries. Seeing that this is the 95 % prediction interval, so we'd expect about 5 % of such countries to be outside that range we just calculated. In summary, the variation between countries with similar tax to GDP ratio dominates.

A few more things to consider / thoughts:

Particular social insurance varies across countries not only in quality and costs, but also in how it is organized. And that latter can vary from 100% in government hand (and tax paid) over mandatory but separate from taxes and available if the individual wants to informal (family + friends) to basically unavailable for normal people. The first 2 possiblities can vary from  0 to 100 % tax money - even for the same percentage of GDP going into health care. For example, health care in Germany is to a large extent not by tax (i.e. government budget), but by compulsory health insurance fees. The extent of this is another 0.07 of GDP. Compare to Italy, where health care is financed by taxes. That is, the Italian tax : GDP ratio includes health care, whereas for Germany that comes on top.
So we do have a correlation. As we all know, that doesn't make a causation. And IMHO we can have plausible causation stories in both directions:

paying taxes makes people unhappy - they have less money which they can spend on what they need most. 
That is, unless they perceive that they get back a sensible value of their taxes: if taxes are spent on things people in fact think important.
This is something I've anecdotally heard e.g. Italians complain about: at 43.5 % of GDP they are taxed about as heavily as Germany - but they don't feel they get as much for their taxes (but see the health care difference above), in particular, they perceive that their government/administration is inefficient (we Germans do complain about that as well, but after spending some years in Italy I think we're in fact much better off)
low taxes can be a sign of various situations a country can be in, ranging from efficiency (Switzerland), i.e. being able to supply service without needing too much taxes to governments that are unable to collect taxes which may be a symptom of general inefficiency. 
Maybe the lower line of happines vs. tax : GDP ratio indicates that if the heavy taxation doesn't lead to happiness, people are unwilling to pay those taxes...

Even if you look at high tax high happiness countries, you could have either:

taxes paid making people happy or
happiness making people pay more taxes (being OK with being taxed)

* regression for the OECD countries only does not yield a significant slope, and the same is also true for the non-OECD countries. But non-OECD countries have on average lower happiness scores and lower tax to GDP ratios.

A:

Analyzing comparative government policies is hard, since there's no way to have a control group or rigorous studies.  All political scientists can do is analyze reported data and try to tease out correlations.
On this subject (as would be expected) there is no definitive answer, but it has been researched.  For example, here is one article which discusses it, based on  The Political Economy of Human Happiness by Benjamin Radcliff.
The article states:

Indeed, the evidence assessed by Radcliff suggests [a link between government spending and happiness]. His data shows that for one of the metrics, linked to welfare spending, countries scoring high on this indicator, happiness levels are above one point higher than low-scoring countries. He suggests that this contribution to happiness is double that of being married (being married is positively correlated with happiness), and three times the negative drag of unemployment.

However, it also includes this graph, which shows a very weak correlation:

The graph here compares taxation levels (tax revenue as % of GDP) with happiness levels (life satisfaction), based on data from the OECD and the World Happiness Report quoted above. It shows an increasing trendline, associating a level of taxation of 20% in this group of OECD countries with a happiness level of around 6.5. All others thing equal, a level of 50% is correlated with a happiness level of around 6.8: some one thirds of a point higher across the trendline.
But not all others things are equal: the distribution is broad and the effects are very diverse. Denmark is on the top right with a happiness level of 7.526 and the very highest tax level of 50,9. On the far left, we find Switzerland with a marginally lower happiness level of 7.509 and only half the tax rate at 26.6%. On the lowest part of the graph, with happiness levels just above 5 points, we find Portugal, Greece and Hungary, with taxation levels around 34-38%.

So there is research on the subject, but no definitive relationship.

