Is it bad to have error bars constructed with standard deviation that spans to the negative scale while the variable itself shouldn't be negative?

No, in this case, it does not make sense to draw error bars using SDs.

Take a step back. Why do we draw error bars with SDs? As you write, it's to show where "much" of the data lies. This makes sense if your data come from a normal distribution: 68% of your data will lie within $pm 1$ SD from the mean, so showing the mean with an error bar of $pm 1$ SD will give you an interval that contains 68% of your data.

However, the number of visits to a doctor is a count, so it is discrete. And it can't be negative. Thus, it can't be normal. For high counts, you can often treat counts as normal, but not for a mean of 3 and an SD of 5. Using SD-based error bars is the wrong way of answering the original question, i.e., showing where "much" of the data falls.

Better: calculate the top and bottom ends of your interval directly, by calculating (e.g.) the 16% and the 84% quantile of your observations. The range between them will again contain 68% of your data, as in the normal case the interval around the mean $pm 1$ SD.

Alternatively, you can fit a distribution. For instance, a mean of 3 and an SD of 5 are consistent with a negative binomial distribution with a mean of 3 and a size parameter of $frac{3^2}{5^2-3}$ (see R's help page ?qnbinom - there are many different parameterizations of the negbin). For such a distribution, we can again calculate the parametric 16%/84% quantiles, which turns out to give us an interval $[0,6]$:

> qnbinom(pnorm(c(-1,1)),mu=3,size=3^2/(5^2-3))
[1] 0 6