How do I calculate the services sector as a percentage of GDP? Could anyone explain how I should calculate the share of services sector as a percentage of GDP? 

I get the numbers for US GDP from Table 3: https://www.bea.gov/system/files/2019-10/gdp3q19_adv_0.pdf

The GDP in 2018 (numbers in billions of $) was 20,580.2 and services accounted for 9,633.9, which gives us 46.8% of total GDP. However, there are many resources that say services make up about 80% of the US economy, including Statista  or Wikipedia (under the 4.1 Nominal GDP sector composition section).
Note that I use the numbers for 2018, Statista only up to 2016, and Wikipedia  for 2015-16, but that shouldn't make so much of a difference.

What am I missing in my calculations?
The table you are looking at is a GDP expenditure measure.  But this ignores the fact that services are involved in the costs of goods purchased, and that goods are involved in the cost of services purchased.  For example, if you buy a cake in a supermarket, the price reflects not just the food manufacturing industry, but also agriculture, transportation, retail trade and many more.

To consider the importance of different industries in the economy, better analysis comes from the GDP income measure considering the value added (essentially wages and profits) for each industry. So looking at table 5 of another BEA release on GDP by industry, and stripping out duplicated values for sub-industries, you get something like  

                                                                     2018
                                                                $ billion
Gross domestic product                                           20,580.2

--Private industries--
Agriculture, forestry, fishing, and hunting                         166.5
Mining                                                              346.6
Utilities                                                           325.9
Construction                                                        839.1
Manufacturing                                                     2,321.2
Wholesale trade                                                   1,212.2
Retail trade                                                      1,126.9
Transportation and warehousing                                      658.1
Information                                                       1,067.7
Finance, insurance, real estate, rental, and leasing              4,301.6
Professional and business services                                2,579.4
Educational services, health care, and social assistance          1,792.5
Arts, entertainment, recreation, accommodation, and food services   860.6
Other services, except government                                   437.2
--Government--
Federal                                                             790.8
State and local                                                   1,753.8


If you add up the first five rows (Agriculture through to Manufacturing) you get \$3,993.3bn (19.4% of total GDP), while if you add up the other rows you get \$16,580.8bn (80.6% of total GDP)