That article doesn't provide references, and so is suspect. Any drug needs to have the effect replicated in multiple independent laboratories to verify efficacy.
We do have some experimental evidence that beetroot juice has some cytotoxicity for several tumour lines but the effect is much less than a standard chemotherapeutic drug such as Adriamycin 
Both doxorubicin and the beetroot extract exhibited a dose-dependent
cytotoxic effect in the two cancer cell lines tested. Although the
cytotoxicity of the beetroot extract was significantly lower when
compared to doxorubicin, it continued to decrease the growth rate of
the PC-3 cells (3.7% in 3 days vs. 12.5% in 7 days) when tested at the
concentration of 29 µg/ml. In contrast, doxorubicin, at the same
concentration level, completely inhibited the growth of the PC-3 cells
in three days. Similarly, comparative studies in the normal human skin
FC and liver HC cell lines showed that the beetroot extract had
significantly lower cytotoxic effect than doxorubicin (8.6% vs. 100%,
respectively, at 29 µg/ml concentration of each, three-day test
Animal experiments suggest that the anti-cancer effect is by inhibition of angiogenesis ( new blood vessel growth ) and induction of apoptosis (programmed cell death), and perhaps like many vegetables, they have a role in cancer prevention.
Betanin reduced the number of CD31+ endothelial microvessels and
increased the expression of caspase-3, suggesting that the lung tumor
inhibitory effects were through induction of apoptosis and inhibition
of angiogenesis. Betanin also induced apoptosis through activated
caspase-3, -7, -9, and PARP in human lung cancer cell lines. Our data
show that betanin significantly inhibits lung tumorigenesis in A/J
mice and merits investigation as a chemopreventive agent for human
lung cancer. 
 Kapadia GJ, Azuine MA, Rao GS, [..], Tokuda H. Cytotoxic effect of the red beetroot (Beta vulgaris L.) extract compared to doxorubicin (Adriamycin) in the human prostate (PC-3) and breast (MCF-7) cancer cell lines. Anticancer Agents Med Chem. 2011 Mar;11(3):280-4. PubMed PMID: 21434853.
 Zhang Q, Pan J, Wang Y, [..], You M. Beetroot red (betanin) inhibits vinyl carbamate- and benzo(a)pyrene-induced lung tumorigenesis through apoptosis. Mol. Carcinog. 2013 Sep;52(9):686-91. doi: 10.1002/mc.21907. PubMed PMID: 22456940.