Why does the standard error of the intercept increase the further $\bar x$ is from 0? The standard error of the intercept term ($\hat{\beta}_0$) in $y=\beta_1x+\beta_0+\varepsilon$ is given by $$SE(\hat{\beta}_0)^2 = \sigma^2\left[\frac{1}{n}+\frac{\bar{x}^2}{\sum_{i=1}^n(x_i-\bar{x})^2}\right]$$
where $\bar{x}$ is the mean of the $x_i$'s.
From what I understand, the SE quantifies your uncertainty- for instance, in 95% of the samples, the interval $[\hat{\beta}_0-2SE,\hat{\beta}_0+2SE]$ will contain the true $\beta_0$. I fail to understand how the SE, a measure of uncertainty, increases with $\bar{x}$. If I simply shift my data, so that $\bar{x}=0$, my uncertainty goes down? That seems unreasonable.
An analogous interpretation is - in the uncentered version of my data, $\hat{\beta}_0$ corresponds to my prediction at $x=0$, while in the centered data, $\hat{\beta}_0$ corresponds to my prediction at $x=\bar{x}$. So does this then mean that my uncertainty about my prediction at $x=0$ is greater than my uncertainty about my prediction at $x=\bar{x}$? That seems unreasonable too, the error $\epsilon$ has the same variance for all values of $x$, so my uncertainty in my predicted values should be the same for all $x$.
There are gaps in my understanding I'm sure. Could somebody help me understand what's going on?
 A: Because the regression line fit by ordinary least squares will necessarily go through the mean of your data (i.e., $(\bar x, \bar y)$)—at least as long as you don't suppress the intercept—uncertainty about the true value of the slope has no effect on the vertical position of the line at the mean of $x$ (i.e., at $\hat y_{\bar x}$).  This translates into less vertical uncertainty at $\bar x$ than you have the further away from $\bar x$ you are.  If the intercept, where $x=0$ is $\bar x$, then this will minimize your uncertainty about the true value of $\beta_0$.  In mathematical terms, this translates into the smallest possible value of the standard error for $\hat\beta_0$.  
Here is a quick example in R:  
set.seed(1)                           # this makes the example exactly reproducible
x0      = rnorm(20, mean=0, sd=1)     # the mean of x varies from 0 to 10
x5      = rnorm(20, mean=5, sd=1)
x10     = rnorm(20, mean=10, sd=1)
y0      = 5 + 1*x0  + rnorm(20)       # all data come from the same  
y5      = 5 + 1*x5  + rnorm(20)       #  data generating process
y10     = 5 + 1*x10 + rnorm(20)
model0  = lm(y0~x0)                   # all models are fit the same way
model5  = lm(y5~x5)
model10 = lm(y10~x10)


This figure is a bit busy, but you can see the data from several different studies where the distribution of $x$ was closer or further from $0$.  The slopes differ a little from study to study, but are largely similar.  (Notice they all go through the circled X that I used to mark $(\bar x, \bar y)$.)  Nonetheless, the uncertainty about the true value of those slopes causes the uncertainty about $\hat y$ to expand the further you get from $\bar x$, meaning that the $SE(\hat\beta_0)$ is very wide for the data that were sampled in the neighborhood of $x=10$, and very narrow for the study in which the data were sampled near $x=0$.  

Edit in response to comment:  Unfortunately, centering your data after you have them will not help you if you want to know the likely $y$ value at some $x$ value $x_\text{new}$.  Instead, you need to center your data collection on the point you care about in the first place.  To understand these issues more fully, it may help you to read my answer here: Linear regression prediction interval.  
