Why GDP series always has an unit root problem?
Whether GDP has a unit root depends on whom you ask, and what they believe about the nature of business cycle. 

The seminal paper that first applied a unit root test (the ADF test) to macroeconomic time series and extracted economic implication is Nelson-Plosser (1982).
They were unable to reject the unit root null hypothesis for GDP and other standard macro series. They offered the following conclusion:

"...macroeconomic models that focus on monetary disturbances as a source of purely transitory fluctuations may
never be successful in explaining a large fraction of output variation and that stochastic
variation due to real factors is an essential element of any model of macroeconomic fluctuations."

These are fighting words in the debate of Real vs. New Keynesian Business Cycles. They interpret their finding as evidence against the latter. Or at least that variation in output is more attributable to permanent real shocks rather than transient shocks to aggregate demand ("monetary disturbances").

Later, however, in a subsequent milestone result Perron (1989) found that a trend-stationary series with a break in its deterministic trend may be mistaken as unit root. In particular, he overturned Nelson-Plosser's non-rejection result for GDP (and other series), after taking into account the 1929 crash and 1973 oil price shock. 

Another possible example of such trend-stationary series with structural break is the CBO GDP data series covering the Great Recession shown in answer by @1muflon1.

Today, the suggestion that, say, US GDP of the last 20 years has a unit root would probably not be too controversial.
In general, the answer you get depends on the data, the unit root test being applied (there are many), and whether the economist has a strong RBC or New Keynesian bias. 

(For example, Nelson and Plosser were already RBC advocates before publishing their finding.)